I wanted to have a single key stroke that would save existing files, or call the file browser.

Here's a key map for Ctrl-S to accomplish that (place in vimrc file):

" If the current buffer has never been saved, it will have no name,
" call the file browser to save it, otherwise just save it.
nnoremap <silent> <C-S> :if expand("%") == ""<CR>browse confirm w<CR>else<CR>confirm w<CR>endif<CR>

Note that in terminal you must somehow disable interpreting <C-s> by the terminal itself. You may try to use an alias for this: